Linear Equations & Graphing

Slope-Intercept Form

y = mx + b
m = slope (rise/run), b = y-intercept (where line crosses y-axis)

y = 2x + 3: slope is 2, crosses y-axis at (0, 3)

Finding Slope

m = (y₂ − y₁) / (x₂ − x₁)

Points (1, 2) and (3, 8): m = (8−2)/(3−1) = 6/2 = 3

Point-Slope Form

y − y₁ = m(x − x₁) — useful when you know a point and slope

Slope 4, passes (2, 1): y − 1 = 4(x − 2) → y = 4x − 7

Standard Form

Ax + By = C where A, B, C are integers. Useful for finding intercepts quickly.
x-intercept: set y = 0. y-intercept: set x = 0.

Parallel & Perpendicular

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals (m₁ × m₂ = −1).

Slope 2 → perpendicular slope = −1/2

Step by Step: Graphing From Slope-Intercept Form

Once an equation is in y = mx + b form, you can draw the line in three moves without a table of values.

Graph y = (2/3)x − 1

Plot the y-intercept first: b is −1, so start at (0, −1).
Use the slope as rise over run: 2/3 means up 2, right 3. That lands you on (3, 1).
Repeat once more for accuracy, then draw the line through your points.

For a negative slope like −2/3, go down 2 and right 3 instead.

What Slope Actually Tells You

Slope is a rate of change: how much y moves for each single step in x. That interpretation is what makes lines useful outside of maths class.

A positive slope rises left to right; a negative slope falls.
A slope of 0 is a flat horizontal line — y never changes.
An undefined slope is a vertical line, because the run is zero and you cannot divide by zero.
A steeper line has a larger absolute slope, regardless of sign.

Slope as a real rate

If y = 15x + 40 describes a repair bill, the slope 15 means each hour of labour adds $15, and the intercept 40 is the call-out fee you pay before any work starts.

Writing the Equation From Two Points

Find the line through (2, 5) and (6, 13)

Slope: m = (13 − 5) / (6 − 2) = 8/4 = 2
Use point-slope with either point: y − 5 = 2(x − 2)
Simplify: y = 2x + 1
Check with the other point: 2(6) + 1 = 13. Correct.

Common Mistakes

Watch out for these

Flipping the slope formula. It is change in y over change in x, not the reverse.
Mixing up the order of the points. Subtract both coordinates in the same order, every time.
Confusing zero slope with undefined slope. Horizontal is 0; vertical is undefined.
Reading b as the x-intercept. It is where the line crosses the y-axis.

Where You Will Use This

Any fixed fee plus a per-unit rate: phone plans, taxi fares, utility bills.
Converting units, such as Celsius to Fahrenheit, which is a straight line.
Reading trends on a science graph and estimating values between measurements.
Budgeting, where a starting balance and a steady rate of spending form a line.

Summary

y = mx + b gives slope and y-intercept at a glance.
Slope is rise over run, and it means a rate of change.
Point-slope form is the fastest route when you know a point and a slope.
Parallel lines share a slope; perpendicular slopes multiply to −1.

FAQ

What does a slope of 0 mean? Horizontal line — no rise.

What is an undefined slope? Vertical line — you can't divide by zero run.

What makes an equation linear? Every variable appears to the first power only. No squares, no square roots, no variables in a denominator.

How do I find intercepts from standard form? Set y to 0 and solve for the x-intercept; set x to 0 and solve for the y-intercept.

Which point do I use in point-slope form? Either one. Both produce the same simplified equation, which makes the second point a free check.

Why can vertical lines not be written as y = mx + b? Because their slope is undefined. They are written as x = a number instead.

Quick Quiz

Test what you just learned. Choose the best answer for each question.